Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree

Hannah Guggiari, Alex Scott
2021 Electronic Journal of Combinatorics  
For every $n\in\mathbb{N}$ and $k\geqslant2$, it is known that every $k$-edge-colouring of the complete graph on $n$ vertices contains a monochromatic connected component of order at least $\frac{n}{k-1}$. For $k\geqslant3$, it is known that the complete graph can be replaced by a graph $G$ with $\delta(G)\geqslant(1-\varepsilon_k)n$ for some constant $\varepsilon_k$. In this paper, we show that the maximum possible value of $\varepsilon_3$ is $\frac16$. This disproves a conjecture of Gyárfas
more » ... ecture of Gyárfas and Sárközy.
doi:10.37236/9039 fatcat:wyb44qoirjbmnkys4nt6d2toge